# Abstract Nonsense

## A Classification of C-representations With No (rho,J)-invariant Subspaces

Point of post: In this post we shall classify, up to equivalence, the set of all $\mathbb{C}$-representations of a finite group $G$. This will enable us, due to a previous characterization, the set of all irreducible $\mathbb{R}$-characters of a finite group $G$ in terms of the irreducible $\mathbb{C}$-characters.

Motivation

Clearly our goal in these last few posts was to see how much information about irreducible $\mathbb{R}$-representations we could glean from our knowledge of irreducible $\mathbb{C}$-representations in the sense that if we know everything we could possibly want to know about the latter for a specific group what could we say about the former. We began this search by showing that there was a bijection between equivalency classes of $\mathbb{C}$-representations $\rho$ satisfying the real condition with realizer $J$ and no non-trivial proper $\left(\rho,J\right)$-invariant subspaces and equivalency classes of  irreducible $\mathbb{R}$-representations where the class $[\rho]$ in the domain was mapped to the class $[\rho_\Re]$ in the codomain. We then showed that there was a natural way to create these special $\mathbb{C}$-representations from irreducible representations which were either complex or quaternionic. What we shall now do in this post is classify all of these special $\mathbb{C}$-representations by showing that, up to equivalence, any such one is either a real irreducible $\mathbb{C}$-representation or of the form of one of the representations created by a complex or quaternionic irrep. This will allow us then, since characters are invariant under equivalence, to ascertain all the irreducible $\mathbb{R}$-characters of the group in terms of the irreducible $\mathbb{C}$-characters.

Classification of $\mathbb{C}$-representations Satisfying the Real Condition With Realizer $J$ and having no non-trivial proper $\left(\rho,J\right)$-invariant Subspaces

For the proof that follows allow $\Delta_G$ to be the set of all $\mathbb{C}$-representations on $G$ satisfying the real condition where, if the realizer is $J$, there does not exist any non-trivial proper $\left(\rho,J\right)$-invariant subspaces. With this in mind:

Theorem: Let $G$ be a finite group and $\rho\in\Delta_G$ with realizer $J$. Then, either $\rho$ is a real irreducible $\mathbb{C}$-representation or there exists a complex or quaternionic irrep $\psi$ and a complex conjugate $K$ such that $\rho\cong\psi\oplus\text{Conj}_\psi^K$ ( in accordance with the construction in the last theorem).

Proof: Since being a representation of $G$ which satisfies the real condition with no non-trivial proper $(\rho,J)$-invariant subspaces we may then assume that $\rho$ is the representation $\displaystyle \psi_1\oplus\cdots\oplus\psi_n:G\to\mathcal{U}\left(\mathscr{V}_1\oplus\cdots\mathscr{V}_n\right)$ where $\psi_k:G\to\mathcal{U}\left(\mathscr{V}_k\right)$ is an irrep for $k=1,\cdots,n$ and $K$ the complex conjugate on $\mathscr{V}_1\oplus\cdots\oplus\mathscr{V}_n$ which is realizer for $\psi_1\oplus\cdots\oplus\psi_n$. Now, if $n=1$ then $\psi_1\oplus\cdots\oplus\psi_n=\psi_1$ and $\psi_1$ is an irrep which must be real since it must commute with $K$ (of course we are appealing to our previous characterization of self-conjugate irreps). Now, suppose that $n\geqslant 2$. We claim that if $\mathscr{H}=\mathscr{V}_1\times\{\bold{0}\}\times\cdots$ then $\mathscr{H}+J\left(\mathscr{H}\right)$ is a $\left(\psi_1\oplus\cdots\oplus\psi_n,J\right)$-invariant subspace. Indeed, this is clear, namely if $(h,0,\cdots)+J((h',0,\cdots))\in\mathscr{H}+J\left(\mathscr{H}\right)$ then for every $g\in G$

\displaystyle \begin{aligned}\left(\psi_1(g)\oplus\psi_n(g)\right)\left((h,0,\cdots)+J\left((h',0,\cdots)\right)\right) &=\left(\left(\psi_1(g)\right)(h),0,\cdots\right)+J\left(\psi_1(g)\oplus\cdots\oplus\psi_n(g)\right)\left(h',0,\cdots,\right)\\ &= \left(\left(\psi_1(g)\right)(h),0,\cdots,\right)+J\left(\left(\psi_(g)\right)(h'),\cdots\right)\end{aligned}

and this last term is clearly in $\mathscr{H}_1+J\left(\mathscr{H}_1\right)$. Since $g\in G$ and $h+J(h')\in\mathscr{H}$ were arbitrary it follows that $\mathscr{H}+J\left(\mathscr{H}\right)$ is $\psi_1\oplus\cdots\psi_n$-invariatn as claimed, and since $\mathscr{H}+J\left(\mathscr{H}\right)$ is evidently $J$-invariant, $\left(\psi_1\oplus\cdots\oplus\psi_n,J\right)$-invariance follows. But, by assumption this implies that $\mathscr{V}_1\oplus\cdots\oplus\mathscr{V}_n=\mathscr{H}+J\left(\mathscr{H}\right)$ from where it follows that $n=1$ if $J\left(\mathscr{H}\right)=\mathscr{H}$ and $n=2$ otherwise. So, we are restricting our attention to representations of the form $\psi_1\oplus\psi_2:G\to\mathcal{U}\left(\mathscr{V}_1\oplus\mathscr{V}_2\right)$. Let, $\mathscr{H}_1=\mathscr{V}_1\times\{\bold{0}\}$ and $\mathscr{H}_2=\{\bold{0}\}\times\mathscr{V}_2$ and $\pi_1,\pi_2$ the projections on these subspaces respectively. Note from the above we’ve determined that $J\left(\mathscr{H}_1\right)\ne\mathscr{H}_1$. Consider then the mapping $\pi_2J\pi_1:\mathscr{V}_1\oplus\mathscr{V}_2\to\mathscr{H}_2$. This clearly must be a non-zero transformation since there exists some $(v,0)\in\mathscr{H}_1$ such that $J(v,0)\notin\mathscr{H}_1$, and so $J(v,0)=(u,w)$ where $w\ne 0$ and so $\pi_2J\pi_1(v,0)=\pi_2J(v,0)=\pi_2(u,w)=w\ne 0$. Moreover, there is an interesting relationship between the map $\pi_2J\pi_1\left(\psi_1(g)\oplus\mathbf{1}\right)$ for each $g\in G$. Namely, for any $(u,v)\in\mathscr{V}_1\oplus\mathscr{V}_2$

\begin{aligned}\pi_2J\pi_1\left(\psi_1(g)\oplus\mathbf{1}\right)(u,v) &=\pi_2J\pi_1\left(\left(\psi_1(g)\right)(u),v\right)\\ &=\pi_2J\left(\left(\left(\psi_1(g)\right)(u),0\right)\right)\\ &=\pi_2J\left(\psi_1(g)\oplus\psi_2(g)\right)(u,0)=\pi_2\left(\psi_1(g)\oplus\psi_2(g)\right)J(u,0)\\ &=\left(\mathbf{1}\oplus\psi_2(g)\right)\pi_2 J(u,0)\\ &= \left(\mathbf{1}\oplus\psi_2(g)\right)\pi_2 J\pi_1(u,v)\end{aligned}

From this, it follows that if $C$ is a complex conjugate on $\mathscr{V}_2$ then we have

$\left(\mathbf{1}\oplus C\psi_2(g)C\right)\left(\mathbf{1}\oplus C\right)\pi_2 J\pi_1=\left(\mathbf{1}\oplus C\right)\pi_2 J\pi_1\psi_1(g)$

Thus, if $T:\mathscr{V}_1\to\mathscr{V}_2$ is given by $v\mapsto (v,0)\mapsto \left(\mathbf{1}\oplus C\right)\pi_2J\pi_1=(u,w)\mapsto w$ then $T$ is an intertwinor for $\psi_1$ and $\text{Conj}_{\psi_2}^{C}$ and since $T$ is non-zero by prior discussion we may conclude by Schur’s lemma that $\psi_1\cong\text{Conj}_{\psi_2}^C$. Thus, $\psi_1\oplus\psi_2\cong\psi_1\oplus \text{Conj}_C^{\psi_1}$. The conclusion follows. $\blacksquare$

References:

1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.